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In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal.41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices. [ABSTRACT FROM AUTHOR]

We create an invariant of virtual Y -oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra. [ABSTRACT FROM AUTHOR]

We define additional gradings on two generalizations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened 3 -manifolds. In particular, the invariants of embedded surfaces are expressed in terms of certain diagrams related to the thickened 3 -manifold, so that we refer to them as picture-valued invariants. This paper contains the first instance of such invariants for 2 -dimensional objects. The additional gradings are defined using cohomological and homotopic information of surfaces: using this information we decorate the smoothings of the standard Khovanov cube, before transferring the decorations into algebra. [ABSTRACT FROM AUTHOR]

We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper. [ABSTRACT FROM AUTHOR]

Square mosaic knots have many applications in algebra, such as modeling quantum states. In this paper, we extend mosaic knot theory to a theory of hexagonal mosaic links, which are links embedded in a plane tiling of regular hexagons. We investigate hexagonal mosaic links created from particular patches of hextiles with a high number of crossings, which we describe as saturated diagrams. Considering patches of varying size and shape, we compute the number of link components that are produced in these saturated diagrams and for special families we identify the knot types of the components. Finally, we discuss open questions relating to saturated diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we define an explicit basis for the 𝔤 𝔩 n -web algebra H n (k) (the 𝔤 𝔩 n generalization of Khovanov's arc algebra) using categorified q -skew Howe duality. Our construction is a 𝔤 𝔩 n -web version of Hu–Mathas' graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and H n (k) , and it gives an explicit graded cellular basis of the 2 -hom space between two 𝔤 𝔩 n -webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky 𝔤 𝔩 n -link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only F. [ABSTRACT FROM AUTHOR]

We compute the action of the 2 -skein algebra of the torus on the 2 -skein module of the solid torus. As a result, we show that the 2 -skein modules of lens spaces is spanned by (⌊ p 2 ⌋ + 1) (2 ⌊ p 2 ⌋ + 1) elements. [ABSTRACT FROM AUTHOR]

In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial Z L of a periodic virtual link L via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math.318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001]. [ABSTRACT FROM AUTHOR]

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern–Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, the first author, and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern–Simons theory where examples are scarce. [ABSTRACT FROM AUTHOR]

The linking set is an invariant of algebraic plane curves introduced by Meilhan and the first author. It has been successfully used to detect several examples of Zariski pairs, i.e. curves with the same combinatorics and different embedding in . Differentiating Shimada's -equivalent Zariski pair by the linking set, we prove, in the present paper, that this invariant is not determined by the fundamental group of the curve. [ABSTRACT FROM AUTHOR]

We show that for any positive integers a,b,m, and n, the Alexander polynomial of the (am,bn)-Turk’s head link is divisible by that of the (m,n)-Turk’s head link. [ABSTRACT FROM AUTHOR]

We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with U (1) × U (1) -equivariant cohomology of ℂ ℙ 1 . Motivated by the relationship between the Temperley–Lieb algebra and annular Khovanov homology, we also introduce an equivariant analog of the Temperley–Lieb algebra. [ABSTRACT FROM AUTHOR]

We prove that the balanced Chekhov–Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov–Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong's quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the SL 2 -character variety. This algebraic morphism shares many resemblances with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative SL 2 character variety. [ABSTRACT FROM AUTHOR]

Let R f = ℤ [ A ± 1 ] be the algebra of Laurent polynomials in the variable A and let R a = ℤ [ A ± 1 , z 1 , z 2 , ... ] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z 1 , z 2 , .... For n ≥ 1 we denote by VB n the virtual braid group on n strands. We define two towers of algebras { VTL n (R f) } n = 1 ∞ and { ATL n (R a) } n = 1 ∞ in terms of diagrams. For each n ≥ 1 we determine presentations for both, VTL n (R f) and ATL n (R a). We determine sequences of homomorphisms { ρ n f : R f [ VB n ] → VTL n (R f) } n = 1 ∞ and { ρ n a : R a [ VB n ] → ATL n (R a) } n = 1 ∞ , we determine Markov traces { T n ′ f : VTL n (R f) → R f } n = 1 ∞ and { T n ′ a : ATL n (R a) → R a } n = 1 ∞ , and we show that the invariants for virtual links obtained from these Markov traces are the f -polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each n ≥ 1 , the standard Temperley–Lieb algebra TL n embeds into both, VTL n (R f) and ATL n (R a) , and that the restrictions to { TL n } n = 1 ∞ of the two Markov traces coincide. [ABSTRACT FROM AUTHOR]

We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group 𝒰 q (𝔰 𝔩 2) explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of 𝒰 q (𝔰 𝔩 2) -invariants (quantum global sections) with the spherical double affine Hecke algebra of type (C 1 ∨ , C 1) , in the four-punctured sphere case, and with the "cyclic deformation" of U (s u 2) in the punctured torus case. In both cases, we give an identification with the corresponding quantum Teichmüller space as proposed by Teschner and Vartanov as a quantization of the moduli space of flat connections. [ABSTRACT FROM AUTHOR]

We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H [ R ] , K [ Q ] and D [ T ] , based on the invariants of knots, R , Q and T , denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R , Q , T) on sublinks of a given link L , obtained by partitioning L into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link. [ABSTRACT FROM AUTHOR]

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces L (p , 1) from the HOMFLYPT skein module of the solid torus, 𝒮 (ST) , it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant X for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set Λ aug , augmenting the basis Λ of 𝒮 (ST). [ABSTRACT FROM AUTHOR]

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this invariant is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of nonisotopic links than the original Jones polynomial, such as the Thistlethwaite link from the unlink with two components. [ABSTRACT FROM AUTHOR]

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra Y is said to be a Frobenius algebra if there exists a Y -module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra Y is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided Y -module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows S 2 -permuted opposite algebra to be extended to S 3 -permuted algebras. [ABSTRACT FROM AUTHOR]

We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open questions. We establish a relation between the Jones period of a knot and the number of sheets of the surfaces that satisfy the Strong Slope Conjecture (Jones surfaces). We also present numerical and experimental evidence supporting a stronger such relation which we state as an open question. [ABSTRACT FROM AUTHOR]

We give an explicit basis of the quotient of the Kauffman bracket skein algebra on a surface by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal. [ABSTRACT FROM AUTHOR]

We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring . This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author's previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov's tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov's conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in where is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov's approach to the Jones polynomial. [ABSTRACT FROM AUTHOR]

Generalizing the construction of the Kauffman bracket skein algebra, Roger and Yang defined the Kauffman bracket arc algebra of a punctured surface, so that it is a quantization of the decorated Teichmüller space. We provide an explicit, finite set of generators for their algebra. [ABSTRACT FROM AUTHOR]

We generalize Ng's two-variable algebraic/combinatorial zeroth framed knot contact homology for framed oriented knots in to knots in , and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ng's three-variable knot contact homology. Our main tool is Lin's generalization of the Markov theorem for braids in to braids in . We conjecture that our framed cord algebras are always finitely generated for non-local knots. [ABSTRACT FROM AUTHOR]

We study the relation between the maximum z-degree of Kauffman polynomial and the determinant for a quasi-alternating link. This gives a stronger criterion for deciding whether a non-alternating link is quasi-alternating or not than the known criterion in terms of Q-polynomials. Also, we determine all quasi-alternating links with determinant 5. [ABSTRACT FROM AUTHOR]

The Jones-Wenzl idempotents are elements of the Temperley-Lieb (TL) planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra (PSPA), which contains the TL planar algebra. It is motivated by Jones' idea of the graph planar algebra (GPA) of type An. In the tensor category of idempotents of the PSPA, the nth Jones-Wenzl idempotent is isomorphic to a direct sum of n + 1 diagrams consisting of only vertical strands. [ABSTRACT FROM AUTHOR]